Heun's method

In mathematics, specifically in operator K-theory, the Baum–Connes conjecture suggests a link between the K-theory of the C*-algebra of a group and the K-homology of the corresponding classifying space of proper actions of that group.

It thus sets up a correspondence between different areas of mathematics, the K-homology being related to geometry, differential operator theory, and homotopy theory, while the K-theory of the reduced $C^{*}$-algebra is a purely analytical object.

The conjecture, if true, would have some older famous conjectures as consequences. For instance, the surjectivity part implies the Kadison–Kaplansky conjecture for a discrete torsion-free group, and the injectivity is closely related to the Novikov conjecture.

The conjecture is also closely related to index theory, as the assembly map $\mu$ is a sort of index, and it plays a major role in Alain Connes' noncommutative geometry program.

The origins of the conjecture go back to Fredholm theory, the Atiyah–Singer index theorem and the interplay of geometry with operator K-theory as expressed in the works of Brown, Douglas and Fillmore, among many other motivating subjects.

Formulation

Let Γ be a second countable locally compact group (for instance a countable discrete group). One can define a morphism

$\mu :R{K}_{*}^{\mathrm{\Gamma }}(\underset{\_}{E\mathrm{\Gamma }})\to K_{*}(C_{\lambda }^{*}(\mathrm{\Gamma })),$

called the assembly map, from the equivariant K-homology with $\mathrm{\Gamma }$-compact supports of the classifying space of proper actions $\underset{\_}{E\mathrm{\Gamma }}$ to the K-theory of the reduced C*-algebra of Γ. The index * can be 0 or 1.

Paul Baum and Alain Connes introduced the following conjecture (1982) about this morphism:

The assembly map μ is an isomorphism.

As the left hand side tends to be more easily accessible than the right hand side, because there are hardly any general structure theorems of the $C^{*}$-algebra, one usually views the conjecture as an "explanation" of the right hand side.

The original formulation of the conjecture was somewhat different, as the notion of equivariant K-homology was not yet common in 1982.

In case $\mathrm{\Gamma }$ is discrete and torsion-free, the left hand side reduces to the non-equivariant K-homology with compact supports of the ordinary classifying space $B\mathrm{\Gamma }$ of $\mathrm{\Gamma }$.

There is also more general form of the conjecture, known as Baum–Connes conjecture with coefficients, where both sides have coefficients in the form of a $C^{*}$-algebra $A$ on which $\mathrm{\Gamma }$ acts by $C^{*}$-automorphisms. It says in KK-language that the assembly map

${\mu }_{A,\mathrm{\Gamma }}:RK{K}_{*}^{\mathrm{\Gamma }}(\underset{\_}{E\mathrm{\Gamma }},A)\to K_{*}(A{⋊}_{\lambda }\mathrm{\Gamma }),$

is an isomorphism, containing the case without coefficients as the case $A=\mathbb{ℂ}$.

However, counterexamples to the conjecture with coefficients were found in 2002 by Nigel Higson, Vincent Lafforgue and George Skandalis, basing on not universally accepted, as of 2008, results of Gromov on expanders in Cayley graphs. Even provided validity of Higson, Lafforgue & Skandalis, conjecture with coefficients remains an active area of research, since it is, not unlike the classical conjecture, often seen as a statement concerning particular groups or class of groups.

Examples

Let $\mathrm{\Gamma }$ be the integers $\mathbb{\mathbb{Z}}$. Then the left hand side is the K-homology of $B\mathbb{\mathbb{Z}}$ which is the circle. The $C^{*}$-algebra of the integers is by the commutative Gelfand–Naimark transform, which reduces to the Fourier transform in this case, isomorphic to the algebra of continuous functions on the circle. So the right hand side is the topological K-theory of the circle. One can then show that the assembly map is KK-theoretic Poincaré duality as defined by Guennadi Kasparov, which is an isomorphism.

Another simple example is given by compact groups. In this case, both sides identify naturally with the complex representation ring $R(K)$ of the group in such a way that the assembly map becomes the identity.

Results

The conjecture without coefficients is still open, although the field has received great attention since 1982. The conjecture is proved for the following classes of groups:

• Discrete subgroups of $SO(n,1)$ and $SU(n,1)$.
• Groups with the Haagerup property, sometimes called a-T-menable groups. These are groups that admit an isometric action on an affine Hilbert space $H$ which is proper in the sense that $\underset{n\to \mathrm{\infty }}{\lim }{g}_n\xi \to \mathrm{\infty }$ for all $\xi \in H$ and all sequences of group elements $g_n$ with $\underset{n\to \mathrm{\infty }}{\lim }{g}_n\to \mathrm{\infty }$. Examples of a-T-menable groups are amenable groups, Coxeter groups, groups acting properly on trees, and groups acting properly on simply connected $CAT(0)$ cubical complexes.
• Groups that admit a finite presentation with only one relation.
• Discrete cocompact subgroups of real Lie groups of real rank 1.
• Cocompact lattices in $SL(3,\mathbb{ℝ})$,$SL(3,\mathbb{ℂ})$ or $SL(3,{\mathbb{\mathbb{Q}}}_p)$. It was a long-standing problem since the first days of the conjecture to expose a single infinite property T-group that satisfies it. However, such a group was given by V. Lafforgue in 1998 as he showed that cocompact lattices in $SL(3,\mathbb{ℝ})$ have the property of rapid decay and thus satisfy the conjecture.
• Gromov hyperbolic groups and their subgroups.
• Among non-discrete groups, the conjecture has been shown in 2003 by J. Chabert, S. Echterhoff and R. Nest for the vast class of all almost connected groups (i. e. groups having a cocompact connected component), and all groups of $k$-rational points of a linear algebraic group over a local field $k$ of characteristic zero (e.g. $k={\mathbb{\mathbb{Q}}}_p$). For the important subclass of real reductive groups, the conjecture had already been shown in 1982 by A. Wassermann.

Injectivity is known for a much larger class of groups thanks to the Dirac-dual-Dirac method. This goes back to ideas of Michael Atiyah and was developed in great generality by Gennadi Kasparov in 1987. Injectivity is known for the following classes:

• Discrete subgroups of connected Lie groups or virtually connected Lie groups.
• Discrete subgroups of p-adic groups.
• Bolic groups (a certain generalization of hyperbolic groups).
• Groups which admit an amenable action on some compact space.

The simplest example of a group for which it is not known whether it satisfies the conjecture is $S{L}_3(\mathbb{\mathbb{Z}})$.

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
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  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

External links

• On the Baum-Connes conjecture by Dmitry Matsnev.